The auxiliary circle of the hyperbola $\mathrm{xy}=9$ in

$$\begin{gathered} Given rec\tan gular hyperbola xy=9 \\ compare with xy=c^2 we get c^2=9 \end{gathered}$$

equation of Auxiliary circle of $\mathrm{xy}={\mathrm{c}}^2$ is ${\mathrm{x}}^2+{\mathrm{y}}^2=2{\mathrm{c}}^2$

So,equation of auxiliary circle is $x^2+y^2=18$